Baire and $\sigma $-Borel characterizations of weakly compact sets in $M(T)$

Abstract

Abstract. Let T be a locally compact Hausdorff space and let M (T ) be the Banach space of all bounded complex Radon measures on T . Let Bo(T) and Bc(T) be the σ-rings generated by the compact G δ subsets and by the compact subsets of T , respectively. The members of Bo(T ) are called Baire sets of T and those of Bc(T ) are called σ-Borel sets of T (since they are precisely the σ-bounded Borel sets of T ). Identifying M (T ) with the Banach space of all Borel regular complex measures on T , in this note we characterize weakly compact subsets A of M (T ) in terms of the Baire and σ-Borel restrictions of the members of A. These characterizations permit us to give a generalization of a theorem of Dieudonné which is stronger and more natural than that given by Grothendieck.

A simple proof of the Grothendieck theorem on the Dieudonné property of $C\textunderscore 0(T)$

“…by Theorem 50.D of Halmos [8] there exists an open Baire set V n such that (2)⇒(12) Let (f n ) be as in (12). Then lim n f n (t) = f (t) exists in [0,1] for each t ∈ T and f is Borel measurable.…”

“…We obtain in Theorem 1 below, excepting four, all the remaining 31 characterizations given in [13] for a continuous linear map u : C 0 (T ) → X to be weakly compact and the Dieudonné property of C 0 (T ) is an immediate consequence of the equivalence of the assertions (1) and (8) or of (1) and (34) in the said theorem. Of course, the present techniques devoid of the use of Theorem 1 of [12] are not powerful enough to obtain all the 35 characterizations given in [13] (see Remark 3). But, on the other hand, the new characterizations (19), (20) and (21) of Theorem 1 below are also deducible by the techniques of [13] if one invokes the equivalence of (i) and (ii) of Proposition 1 of [13].…”

“…In this note we dispense with the use of Theorems 1 and 2 of [12] and give a simple proof of the Grothendieck theorem on the Dieudonné property of C 0 (T ). The present proof is based on Lemma 1 and Theorem 2 of [7], the first part of Theorem 1 of [13] and the theorem on regular Borel extension of X-valued Baire measures on T .…”

“…Employing new techniques, we not only obtained in [13] the locally compact analogue of Theorem 6 of [7] (which establishes the Dieudonné property of C 0 (T )), but also provided several new characterizations for a continuous linear map u : C 0 (T ) → X to be weakly compact. The proof given in [13] makes use of Lemma 1 and Theorem 2 of [7], Theorem 1 of [13] (which provides a technique, similar to that of Bartle-Dunford-Schwartz in [1], for the lcHs case) and Theorems 1 and 2 of [12], which characterize relatively weakly compact sets in M (T ) in terms of the Baire, σ-Borel and Borel restrictions of the members of the set in question. The proof of Theorem 1 of [12] is quite involved and deep.…”

“…The proof given in [13] makes use of Lemma 1 and Theorem 2 of [7], Theorem 1 of [13] (which provides a technique, similar to that of Bartle-Dunford-Schwartz in [1], for the lcHs case) and Theorems 1 and 2 of [12], which characterize relatively weakly compact sets in M (T ) in terms of the Baire, σ-Borel and Borel restrictions of the members of the set in question. The proof of Theorem 1 of [12] is quite involved and deep.…”